A multigraph is a graph in which we allow multiple edges between two fixed vertices. Formally, a multigraph is a pair where is a (usually finite) set of vertices and is a (also usually finite) multiset of pairs of elements of . Frequently in the context of multigraphs one considers graphs where loops are allowed, i.e. is allowed to contain (possibly several copies of) the pair for some .
Multigraphs may be thought of as special cases of (edge-)weighted graphs, where the weight on each edge is the multiplicity of the edge in . In this interpretation, multigraphs are exactly those weighted graphs for which each edge weight is a positive (or nonnegative) integer.
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